ELT Linear Algebra

TL;DR

This paper introduces ELT algebra, extending tropical algebra with layers, and establishes fundamental properties linking determinants, linear dependence, and ranks, bridging classical and tropical algebraic results.

Contribution

It proves key properties of ELT matrices, including singularity criteria, rank equivalences, and inner product concepts, connecting ELT and Kapranov ranks.

Findings

ELT matrix singularity iff rows are linearly dependent

ELT row rank equals submatrix rank

ELT inner products satisfy Cauchy-Schwarz inequality

Abstract

Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove analogous tropical results. Specifically we study the connection between the ELT determinant and linear dependency, and use a generalized version of Kapranov Theorem proved in [7] (called the Fundamental Theorem). In this paper we prove that an ELT matrix is singular if and only if its rows are linearly dependent and that the row rank and submatrix rank of an ELT matrix are equal. We also define an ELT rank for a tropical matrix, and prove that it is equal to its Kapranov rank. In addition, we formalize the concept of ELT inner products, and prove ELT versions of some known theorems such as Cauchy-Schwarz inequality.